By the resolution of the Willmore conjecture [1]}, \(\mathcal {M}\mathcal {C}^{*}_{3}(\Lambda _\mathcal {C}) = \emptyset \)  for
\(\Lambda _\mathcal {C}= \frac{2\pi ^2}{4\pi } \approx 1.57 > \lambda (\mathbb {S}^1) \approx 1.52.\) 
Thus \((\dagger _{3,\Lambda })\)  holds for all \(\Lambda \le \Lambda _\mathcal {C}\) . Furthermore, by the classification of low-entropy shrinkers in \(\mathbb {R}^3\)  from [2]}, there is \(\varepsilon _0 >0\)  so that \(\mathcal {S}^*_2(\lambda (\mathbb {S}^1) + \varepsilon _0) = \mathcal {S}^\textrm {gen}_2\) . Taking \(\varepsilon _0\)  so that \(\lambda (\mathbb {S}^1) + \varepsilon _0 \le \Lambda _\mathcal {C}\)  the both hypothesis hold with \(n=3\)  and \(\Lambda = \lambda (\mathbb {S}^1) + \varepsilon _0\) .
If \(\lambda (M) \le \lambda (\mathbb {S}^1\times \mathbb {R})\)  then the resolution of the mean convex neighborhood conjecture for \(\mathbb {S}^2\times \mathbb {R}\)  singularities in [3]} implies the non-fattening of the level set flow.
